Finite Group Databasebeta

← #46848193

Subgroups — #46848193

Order 34688.

408 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774543C211yesmaximal, characteristic
2#31774254C211yesmaximal, characteristic, complemented
3#31774227C211yesmaximal, characteristic, complemented
4#31774221C211yesmaximal, characteristic, complemented
5#31774194C211yesmaximal, characteristic, complemented
6#31774136C211yesmaximal, characteristic, complemented
7#31774128C211yesmaximal, characteristic, complemented
8#31774119C211yesmaximal, characteristic, complemented
9#31774057C211yesmaximal, characteristic, complemented
10#31774047C211yesmaximal, characteristic, complemented
11C4×#16864641C211yesmaximal, characteristic, complemented
12D4×C271⋊C8C211yesmaximal, characteristic, complemented
13#31773891C211yesmaximal, characteristic, complemented
14C2×#22576080C211yesmaximal, characteristic, complemented
15#552×C271C211yesmaximal, characteristic, complemented
16C2×#16864654C2211yescharacteristic
17C22×D1084C2211yescharacteristic
18C2×#16864644C2211yescharacteristic, complemented
19C2×#16864641C2211yescharacteristic
20C2×#16864641C2211yescharacteristic
21C22×C271⋊C8C2211yescharacteristic
22C22×C271⋊C8C2211yescharacteristic, complemented
23#22576148C2211yescharacteristic
24#22576145—21no—
25#22576139C2211yescharacteristic
26#22576134C2211yescharacteristic
27#22576133—21no—
28#22576128—21no—
29C4×D1084C2211yescharacteristic
30C22⋊C4×D271—21no—
31#22576093C2211yescharacteristic, complemented
32#22576091C2211yescharacteristic, complemented
33#22576091C2211yescharacteristic, complemented
34C4⋊(C271⋊C8)C2211yescharacteristic, complemented
35#22576084C2211yescharacteristic, complemented
36#22576083C2211yescharacteristic, complemented
37#22576083C2211yescharacteristic, complemented
38#22576083C2211yescharacteristic, complemented
39#22576080C2211yescharacteristic
40#22576080C2211yescharacteristic
41#22576080C2211yescharacteristic
42#22576080C2211yescharacteristic
43#22576080C2211yescharacteristic
44#22576080C2211yescharacteristic
45C22⋊(C271⋊C8)C2211yescharacteristic
46C22⋊(C271⋊C8)C2211yescharacteristic, complemented
47C4×C271⋊C8C2211yescharacteristic, complemented
48#126×C271—21no—
49#125×C271—21no—
50#125×C271C2211yescharacteristic, complemented
51#123×C271—21no—
52C22⋊Q8×C271C2211yescharacteristic
53C22⋊Q8×C271C2211yescharacteristic, complemented
54C4×Q8×C271C2211yescharacteristic, complemented
55C4×D4×C271C2211yescharacteristic
56#117×C271—21no—
57#117×C271C2211yescharacteristic, complemented
58C2×C4⋊C4×C271C2211yescharacteristic
59C23×D271—21no—
60#16864654—22no—
61#16864654—41no—
62C2×Dic542C2311yescharacteristic
63C2×D1084—22no—
64C2×D1084—21no—
65C2×D1084—41no—
66C2×D1084C2311yescharacteristic
67C2×D1084C2311yescharacteristic
68C2×C22⋊D271—21no—
69C2×C22⋊D271—21no—
70C2×C4×D271—21no—
71C2×C4×D271—21no—
72#16864644—41no—
73#16864641—22no—
74#16864641—41no—
75C2×C271⋊C8—41no—
76C2×C271⋊C8—41no—
77C2×C271⋊C8C2311yescharacteristic
78C2×C271⋊C8C2311yescharacteristic
79C2×C271⋊C8C2311yescharacteristic
80C2×C271⋊C8C2311yescharacteristic
81C4⋊Dic271C2311yescharacteristic
82Dic271⋊C4—21no—
83C22⋊Dic271—21no—
84(C2×D271)⋊C4—21no—
85(C2×D271)⋊C4—21no—
86(C2×D271)⋊C4—21no—
87(C2×D271)⋊C4—21no—
88C2×Q8×C271C2311yescharacteristic
89C2×D4×C271D411yescharacteristic
90C22×C4×C271—21no—
91C22×C4×C271C2311yescharacteristic
92C22×C4×C271C2311yescharacteristic
93C4⋊C4×C271—21no—
94C4⋊C4×C271—21no—
95C4⋊C4×C271—21no—
96C4⋊C4×C271—21no—
97C4⋊C4×C271—21no—
98C4⋊C4×C271—21no—
99C4⋊C4×C271D411yescharacteristic
100C4⋊C4×C271C2311yescharacteristic
101C4⋊C4×C271C2311yescharacteristic
102C4⋊C4×C271C2311yescharacteristic
103C22⋊C4×C271—21no—
104C22⋊C4×C271—21no—
105C22⋊C4×C271—21no—
106C22⋊C4×C271D411yescharacteristic
107C22⋊C4×C271D411yescharacteristic
108C42×C271—21no—
109C42×C271—21no—
110C42×C271C2311yescharacteristic
111C22×D271—21no—
112C22×D271—21no—
113C22×D271—42no—
114C22×D271—42no—
115C22×D271—81no—
116Dic542—21no—
117Dic542—21no—
118D1084—21no—
119D1084—21no—
120D1084—21no—
121D1084—42no—
122D1084#4112yes—
123C22⋊D271—42no—
124C22⋊D271—81no—
125C2×Dic271—21no—
126C2×Dic271—21no—
127C4×D271—42no—
128C4×D271—81no—
129C271⋊C8—21no—
130C271⋊C8—21no—
131C271⋊C8—21no—
132C271⋊C8—21no—
133C23×C271C2×D411yescharacteristic
134C23×C271C2×D411yescharacteristic
135Q8×C271—21no—
136Q8×C271—21no—
137D4×C271—22no—
138C2×C4×C271—21no—
139C2×C4×C271—21no—
140C2×C4×C271—21no—
141C2×C4×C271—21no—
142C2×C4×C271—21no—
143C2×C4×C271—21no—
144C2×C4×C271—21no—
145C2×C4×C271—21no—
146C2×C4×C271—42no—
147C2×C4×C271C2×D411yescharacteristic
148C2×C4×C271C2×D411yescharacteristic
149C2×C4×C271C2×D411yescharacteristic
150C2×C4×C271C2×D411yescharacteristic
151C2×C4×C271C2411yescharacteristic
152C22×C271—21no—
153C22×C271—21no—
154C22×C271—21no—
155C22×C271—21no—
156C22×C271—21no—
157C22×C271#13512yes—
158C22×C271C22×D411yescharacteristic
159C2×D271—22no—
160C2×D271—41no—
161C2×D271—41no—
162C2×D271—41no—
163C2×D271—41no—
164C2×D271—41no—
165C2×D271—84no—
166C4×C271—21no—
167C4×C271—21no—
168C4×C271—21no—
169C4×C271—21no—
170C4×C271—41no—
171C4×C271—41no—
172C4×C271—41no—
173C4×C271—41no—
174C4×C271C2×#4111yescharacteristic
175C4×C271H(5,𝔽2)11yescharacteristic
176Dic271—41no—
177Dic271—41no—
178C2×C271—22no—
179C2×C271—41no—
180C2×C271#54611yescharacteristic
181C2×C271C2×#13511yescharacteristic
182C2×C271#57711yescharacteristic
183D271—42no—
184D271—81no—
185C271#329611yescharacteristic, complemented, Sylow
186#3296—2711nomaximal, Sylow
187#552D27111yescharacteristic, complemented
188#546—2711no—
189#488—2711no—
190#483—2711no—
191#482—2711no—
192#478—2711no—
193#467—2711no—
194#465—2711no—
195#464—2711no—
196#452—2711no—
197#450—2711no—
198C4×SD16—2711no—
199C8×D4—2711no—
200#416—2711no—
201C2×#102—2711no—
202C2×#41—2711no—
203C22×D4—2711no—
204C2×Q16—2711no—
205C2×SD16—2711no—
206C2×SD16—2711no—
207C22×C8—2711no—
208C22×C8—2711no—
209#126—21no—
210#125—21no—
211#125C2×D27111yescharacteristic, complemented
212#124—2711no—
213#123—21no—
214#123—5421no—
215C22⋊Q8—2711no—
216C22⋊Q8C2×D27111yescharacteristic
217C22⋊Q8C2×D27111yescharacteristic, complemented
218#121—2711no—
219#121—5421no—
220#120—5421no—
221C4×Q8C2×D27111yescharacteristic, complemented
222C4×D4—2711no—
223C4×D4C2×D27111yescharacteristic
224#117—21no—
225#117C2×D27111yescharacteristic, complemented
226C2×C4⋊C4C2×D27111yescharacteristic
227C2×C22⋊C4—5421no—
228#107—2711no—
229#106—2711no—
230#106—2711no—
231C4⋊C8—2711no—
232#103—2711no—
233#103—2711no—
234#103—2711no—
235#103—2711no—
236#102—2711no—
237#102—2711no—
238#102—2711no—
239#102—2711no—
240#102—2711no—
241#102—2711no—
242C22⋊C8—2711no—
243C22⋊C8—2711no—
244C4×C8—2711no—
245C24—5421no—
246#41—5422no—
247#41—10841no—
248C2×Q8—2711no—
249C2×Q8C22×D27111yescharacteristic
250C2×D4—2711no—
251C2×D4—2711no—
252C2×D4—5422no—
253C2×D4—5421no—
254C2×D4—5421no—
255C2×D4—5421no—
256C2×D4—10841no—
257C2×D4C22⋊D27111yescharacteristic
258C22×C4—21no—
259C22×C4—5421no—
260C22×C4—5421no—
261C22×C4C22×D27111yescharacteristic
262C22×C4C22×D27111yescharacteristic
263Q16—10841no—
264SD16—5422no—
265SD16—10841no—
266C2×C8—2711no—
267C2×C8—2711no—
268C2×C8—2711no—
269C2×C8—2711no—
270C2×C8—10841no—
271C2×C8—10841no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—2711no—
279C4⋊C4—5421no—
280C4⋊C4C22⋊D27111yescharacteristic
281C4⋊C4C22×D27111yescharacteristic
282C4⋊C4C22×D27111yescharacteristic
283C4⋊C4C22×D27111yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—5421no—
288C22⋊C4—5421no—
289C22⋊C4—5421no—
290C22⋊C4—5421no—
291C22⋊C4—5421no—
292C22⋊C4C22⋊D27111yescharacteristic
293C22⋊C4C22⋊D27111yescharacteristic
294C42—21no—
295C42—21no—
296C42C22×D27111yescharacteristic
297C23—5421no—
298C23—5421no—
299C23—10842no—
300C23—10842no—
301C23—21681no—
302C23C2×C22⋊D27111yescharacteristic
303C23C2×C22⋊D27111yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—5421no—
307Q8—5421no—
308D4—22no—
309D4—2712no—
310D4—5421no—
311D4—5421no—
312D4—5421no—
313D4—10842no—
314D4—10842no—
315D4—21681no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—42no—
325C2×C4—5421no—
326C2×C4—5421no—
327C2×C4—10842no—
328C2×C4—21681no—
329C2×C4C2×C22⋊D27111yescharacteristic
330C2×C4C2×C22⋊D27111yescharacteristic
331C2×C4C2×C22⋊D27111yescharacteristic
332C2×C4C2×C22⋊D27111yescharacteristic
333C2×C4C23×D27111yescharacteristic
334C8—5421no—
335C8—5421no—
336C8—5421no—
337C8—5421no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—5422no—
344C22—10841no—
345C22—10841no—
346C22—10841no—
347C22—10841no—
348C22—10841no—
349C22—21684no—
350C22#2257618012yes—
351C22C22×C22⋊D27111yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—10841no—
361C4—10841no—
362C4#41×D27111yescharacteristic
363C4#2257620611yescharacteristic
364C2—22no—
365C2—41no—
366C2—10842no—
367C2—21681no—
368C2#3177453511yescharacteristic
369C2C2×#2257618011yescharacteristic
370C2#3177470511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#552×C271C2
socleC22×C271C22×D4
prime radical#31774543C2
hypercenter#552D271
hypocenterC271#3296
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C4×D4 ◁ #552

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #31774254C2semidirect
3. #31774227C2semidirect
4. #31774221C2semidirect
5. #31774194C2semidirect
6. #31774136C2semidirect
7. #31774128C2semidirect
8. #31774119C2semidirect
9. #31774057C2semidirect
10. #31774047C2semidirect
11. C4×#16864641C2semidirect
12. D4×C271⋊C8C2semidirect
13. #31773891C2semidirect
14. C2×#22576080C2semidirect
15. #552×C271C2semidirect
18. C2×#16864644C22semidirect
22. C22×C271⋊C8C22semidirect
31. #22576093C22semidirect
32. #22576091C22semidirect
33. #22576091C22semidirect
34. C4⋊(C271⋊C8)C22semidirect
35. #22576084C22semidirect
36. #22576083C22semidirect
37. #22576083C22semidirect
38. #22576083C22semidirect
46. C22⋊(C271⋊C8)C22semidirect
47. C4×C271⋊C8C22semidirect
50. #125×C271C22semidirect
53. C22⋊Q8×C271C22semidirect
54. C4×Q8×C271C22semidirect
57. #117×C271C22semidirect
185. C271#3296semidirect
187. #552D271semidirect
211. #125C2×D271semidirect
217. C22⋊Q8C2×D271semidirect
221. C4×Q8C2×D271semidirect
225. #117C2×D271semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#317745432
#317742542
#317742272
#317742212
#317741942
#317741362
#317741282
#317741192
#317740572
#317740472
C4×#168646412
D4×C271⋊C82
#317738912
C2×#225760802
#552×C2712
#3296271